Question

If a function defined by \( f(x) = \frac{(3^x - 1)^2}{\sin x \cdot \log(1+x)}, x \neq 0 \) is continuous at \(x = 0\), then \(f(0) =\) ?

• A. \(2 \log 3\)
• B. \(\log 3^2\)
• C. \(2 + \log 3\)
• D. \((\log 3)^2\)

Hint:

Use Taylor approximations for exponential, logarithmic, and trigonometric functions when evaluating limits.

Answer 1

The Correct Option is D

We are given a function continuous at \(x = 0\), so: \[ f(0) = \lim_{x \to 0} \frac{(3^x - 1)^2}{\sin x \cdot \log(1+x)} \] Using expansions: \[ 3^x - 1 \approx x \log 3,\quad \sin x \approx x,\quad \log(1+x) \approx x \] So: \[ f(x) \approx \frac{(x \log 3)^2}{x \cdot x} = \frac{x^2 (\log 3)^2}{x^2} = (\log 3)^2 \] Therefore, \(f(0) = (\log 3)^2\)